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  "body": "\n## Definition\n\nLet $Y$ be a closed oriented [[3-manifold|R20TB2KEQQ]].\nLet $(\\Sigma, \\alpha, \\beta, z)$ be a pointed Heegaard diagram of $Y$ coming from a [[Heegaard splitting|EDE9919B5R]].\nHere $\\Sigma$ is a closed oriented surface of genus $g$.\nThe families $\\alpha = (\\alpha_1, \\ldots, \\alpha_g)$ and $\\beta = (\\beta_1, \\ldots, \\beta_g)$ consist of disjoint simple closed curves on $\\Sigma$.\nThe base point $z \\in \\Sigma$ lies off all of these curves.\nAssume the diagram is admissible, and fix a generic path of almost complex structures on the symmetric product $\\mathrm{Sym}^g(\\Sigma)$.\n\n1. The tori $\\mathbb{T}_\\alpha = \\alpha_1 \\times \\cdots \\times \\alpha_g$ and $\\mathbb{T}_\\beta = \\beta_1 \\times \\cdots \\times \\beta_g$ are submanifolds of $\\mathrm{Sym}^g(\\Sigma)$ that meet transversally in finitely many points.\n2. The chain complex $\\widehat{CF}(\\Sigma, \\alpha, \\beta, z)$ is the vector space over $\\F_2$ with basis $\\mathbb{T}_\\alpha \\cap \\mathbb{T}_\\beta$.\n3. Let $x$ and $y$ be basis elements.\nConsider the holomorphic disks from $x$ to $y$ with Maslov index $1$ whose domain avoids $z$.\nThe coefficient of $y$ in $\\partial x$ is their number modulo $2$.\nDisks that differ by a translation of the parameter are counted once.\n\nThe homology $\\widehat{HF}(Y)$ of this complex is the *hat version of Heegaard Floer homology* of $Y$.\nThe versions $HF^-$, $HF^\\infty$ and $HF^+$ are defined in the same way over $\\F_2[U]$ or $\\F_2[U, U^{-1}]$.\nIn them the disks may cover $z$, and the power of $U$ records how many times they do.\n\n## Remarks\n\nOzsváth and Szabó proved that $\\partial^2 = 0$ and that the homology does not depend, up to isomorphism, on the diagram and the other choices, so it is an invariant of $Y$.\nFor $Y = S^3$, the genus $1$ diagram with $\\alpha$ and $\\beta$ meeting in one point gives $\\widehat{HF}(S^3) \\cong \\F_2$.\nFor a rational homology sphere $Y$, the dimension of $\\widehat{HF}(Y)$ is at least the order of $H_1(Y; \\Z)$; the spaces with equality are called L-spaces, and lens spaces are examples.\nWith suitable orientation choices the construction also works over $\\Z$.\nIt is a form of Lagrangian [[Floer homology|NCKFPAGHXF]] for the pair $(\\mathbb{T}_\\alpha, \\mathbb{T}_\\beta)$.\n",
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